Subject: CCL:UHF-CISD

From: [email-masked] (Ron Shepard)
Date: Wed, 17 May 95 19:17:30 CDT

In a previous posting, Mike Frisch wrote:

>While we're on the subject, I'd like to clear up a related
>misconception which I've seen in the literature.  The UCISD
>wavefunction in Gaussian is NOT an exact spin eigenfunction.  It
>consists of all determinants produced by replacing one or two
>spin-orbitals which are occupied in the reference determinant with
>virtuals.  For an open-shell system, this would not lead to a
>spin-eigenfunction even if the reference determinant had the same
>alpha and beta orbitals, because some determinants which are a double
>excitation over spacial orbitals but a triple excitation over
>spin-orbitals (because the spins are flipped) are needed to form
>proper spin-eigenfunctions.
>
>To show a simpler example with singles vs. doubles, if the reference
>determinant is 1a 1b 2a, then the determinant 1a 2a 3b is considered a
>single excitation over spin-orbitals but 1a 2b 3a, which should be
>mixed with 1a 2a 3b to form a spin-eigenfunction, is a double
>excitation of spin-orbitals from the reference.  Of course, both of
>these would be included in a CISD wavefunction, but the same issue
>occurs with double and triple excitations.

Whether a CI wave function ends up being a spin eigenfunction depends on
the hamiltonian operator (or approximations thereof), the expansion
space, and possibly the iterative procedure used to converge the
eigenvalue equation.  Mike Frisch was discussing the expansion space
above, but perhaps was not clear on a couple of points.

First, if a reference determinant is |1a1b2a> with the same spatial
orbitals used for different spins (i.e. not UHF), then this reference
determinant is a pure doublet (it is a spin eigenfunction of both Sz and
S^2 operators).  The usual hamiltonian operator (nonrelativistic, no
spin-orbit, etc.)  commutes with both Sz and S^2, so H|ref> will be also
a pure doublet, H^2|ref> will be a pure doublet, and so on for the entire
krylov sequence H^n|ref>.  This applies also if the vectors are truncated
to within some given expansion space.  So if the converged CI wave
function is computed as a linear combination of such krylov vectors, each
of which is a pure doublet, then the final result will also be a pure
doublet.  This argument holds for any sequence of expansion vectors,
truncated or not, provided the operators (and/or methods) used to
generate them commute with Sz and S^2.  And of course this holds also for
any |ref> that is a spin eigenfunction, not just doublets.

In the above example, the spin-adapted excitation operator E(3,1) acting
on |1a1b2a> will give the combination

        |1b2a3a>-|1a2a3b>.  

The other determinant with the same spatial orbital occupations,
|1a2b3a>, is a double spinorbital excitation, as noted above.  It is not,
by itself a pure doublet, but must be combined with the (+) combination
of the other two determinants above to become one.  The correct
combination is

        2*|1a2b3a> - |1b2a3a> - |1a2a3b> 

Some experimentation will show that this spin function cannot be
generated from the reference function |1a1b2a> with any kind of
spin-adapted SINGLE excitation, but it can be generated with a
spin-adapted DOUBLE excitation operator (e.g. the product E(3,2)E(2,1)
results in a linear combination of the above two spin functions).  As
Mike said in his comment, the same thing happens for certain double
spatial orbital excitations that end up being triple spin-orbital
excitations; these do not interact through the hamiltonian with the
original reference function since the hamiltonian is only a spin-adapted
double-excitation operator.

It is possible to compute CI wave functions that consist only of
interacting expansion terms.  It does not matter if these wave functions
are computed in terms of spin-adapted CSFs, or directly in terms of the
determinants.  There are, of course, both advantages and disadvantages to
this approach in practice.  For example, in the COLUMBUS MR-CI code, we
use spin-adapted expansion terms (CSFs), but we can easily discard from
the expansion space only those noninteracting CSFs that involve
excitations from orbitals that are doubly occupied in all reference CSFs.
The other possible non-interacting determinants involving partially
occupied active orbitals are mixed together with the interacting ones.

By the way, the thought might have occurred to someone that there are
only two independent linear combinations of the three determinants
discussed above.  What happened to the other linearly independent
combination, namely |1a2b3a> + |1b2a3a> + |1a2a3b>?  The answer is that
this is also a spin eigenfunction, but it is not a doublet.  It is the
Ms=1/2 component of the 3-electron quartet state, the high-spin component
of which looks like |1a2a3a>.

This finally brings up the question of whether you want a wave function
that is formed only from the interacting space, or if you want all
possible spatial-orbital excitations of a given excitation level included
in the expansion space.  The answer depends on what you want to do with
the wave function.  If you want to compare results of various spatial and
spin symmetries in the most unbiased way possible, or to compute
transition properties between such states, then it is often the case that
the best way to achieve this is to ignore the interacting space
simplifications.

And I won't even try to discuss whether this is ab initio or
semiempirical. ;-)

-Ron Shepard